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We classify the finitely presented restricted Lie algebras among the finitely generated restricted Lie algebras L that are split extension of abelian-by-abelian restricted Lie algebras.

We define and study a pro-p version of Sidki’s weak commutativity construction. This is the pro-p group $$\mathfrak {X}_p(G)$$ generated by two copies G and $$G^{\psi }$$ of a pro-p group, subject to the defining relators $$[g,g^{\psi }]$$ for all $$g \in G$$ . We show for instance that if G is finitely presented or analytic pro-p, then $$\mathfrak {X}_p(G)$$ has the same property. Furthermore we study properties of the non-abelian tensor product and the pro-p version of Rocco’s construction $$\nu (H)$$ . We also study finiteness properties of subdirect products of pro-p groups. In particular we prove a pro-p version of the $$(n-1)-n-(n+1)$$ Theorem.

We calculate the Bieri–Neumann–Strebel–Renz invariant Σ1(G) for finitely presented residually free groups G and show that its complement in the character sphere S(G) is a finite union of finite intersections of closed sub-spheres in S(G). Furthermore, we find some restrictions on the higher-dimensional homological invariants Σn(G, ℤ) and show for the discrete points Σ2(G)dis, Σ2(G, ℤ)dis and Σ2(G, ℚ)dis in Σ2(G), Σ2(G, ℤ) and Σ2(G, ℚ) that we have the equality Σ2(G)dis = Σ2(G, ℤ)dis = Σ2(G, ℚ)dis.

We show that there are uncountably many non-commensurable metabelian pro-[Formula: see text] groups of homological type [Formula: see text] but not of type [Formula: see text], generated by [Formula: see text] element, where [Formula: see text] and [Formula: see text]. In particular there are uncountably many non-commensurable finitely presented metabelian pro-[Formula: see text] groups that are not of type [Formula: see text]. We show furthermore that there are uncountably many non-isomorphic one related 2-generated pro-[Formula: see text] groups.

We construct new classes of self-similar groups: S-arithmetic groups, affine groups and metabelian groups. Most of the soluble ones are finitely presented and of type $$FP_n$$ for appropriate n.

We consider results on finite presentability and homological finiteness property $$FP_n$$ of Lie algebras.

We develop a version of the Bass-Serre theory for Lie algebras (over a field $k$) via a homological approach. We define the notion of fundamental Lie algebra of a graph of Lie algebras and show that this construction yields Mayer-Vietoris sequences. We extend some well known results in group theory to $\mathbb{N}$-graded Lie algebras: for example, we show that one relator $\mathbb{N}$-graded Lie algebras are iterated HNN extensions with free bases which can be used for cohomology computations and apply the Mayer-Vietoris sequence to give some results about coherence of Lie algebras., 27 pages

We calculate the Bieri-Neumann-Strebel-Renz invariant $\Sigma^1(G)$ for even Artin groups $G$ with underlying graph $\Gamma$ such that if there is a closed reduced path in $\Gamma$ with all labels bigger than 2 then the length of such path is always odd. We show that $\Sigma^1(G)^c$ is a rationally defined spherical polyhedron.

We consider the group $\mathfrak{X}(G)$ obtained from $G\ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. Deceptively complicated finitely presented groups arise from this construction: $\mathfrak{X}(G)$ is finitely presented if and only if $G$ is finitely presented, but if $F$ is a non-abelian free group of finite rank then $\mathfrak{X}(F)$ has a subgroup of finite index whose third homology is not finitely generated.

We study the homological finiteness property FPn of fibre products

For a group G and an isomorphism of groups \(\psi : G \rightarrow G^{\psi }\) we study homological and homotopical properties of the group $$\begin{aligned} {\mathcal {E}} (G) = \langle G, G^{\psi } \mid [{\mathcal {D}},{\mathcal {L}}] = 1 \rangle , \end{aligned}$$ where \({\mathcal {D}} = [G, G^{\psi }]\) and \({\mathcal {L}} = [G, \psi ]\). For \(k \ge 2\) we show that \({\mathcal {E}}(G)\) has homotopical type \(F_k\) (resp. homological type \(FP_k\)) if and only if \(G{/}G'\) is finite and G has homotopical type \(F_k\) (resp. homological type \(FP_k\)). In particular \({\mathcal {E}}(G)\) is finitely presented if and only if \(G{/}G'\) is finite and G is finitely presented.

For a group G we study homological and homotopical properties of the group χ ( G ) = 〈 G , G ψ | [ g , g ψ ] = 1 for g ∈ G 〉 . In particular, we show that the operator χ preserves the soluble of type F P ∞ property.

We study when an abstract finitely presented group [Formula: see text] of cohomological dimension [Formula: see text] has pro-[Formula: see text] completion [Formula: see text] of cohomological dimension [Formula: see text]. Furthermore, we prove that for a tree hyperbolic limit group [Formula: see text] we have [Formula: see text] and show an example of a hyperbolic limit group [Formula: see text] that is not free and [Formula: see text] is free pro-[Formula: see text]. For a finitely generated residually free group [Formula: see text] that is not a limit group, we show that [Formula: see text] is not free pro-[Formula: see text].

Let 1 → A → G → Q → 1 be an exact sequence of groups, where A is abelian, Q is polycyclic and ⨂ Q k ( A ⊗ Z Q ) is finitely generated as Q Q -module via the diagonal Q-action for k ≤ 2 m . Moreover we assume that if G is not metabelian, then it is of type FP 3 . Our main result is that sup U ∈ A ⁡ dim Q ⁡ H j ( U , Q ) ∞ for 0 ≤ j ≤ m , where A is the set of all subgroups of finite index in G.

We introduce the notion of virtual endomorphisms of Lie algebras and use it as an approach for constructing self-similarity of Lie algebras. This is done in particular for a class of metabelian Lie algebras having homological type $$FP_n$$ , which are Lie algebra analogues of lamplighter groups. We establish several criteria when the existence of virtual endomorphism implies a self-similar Lie structure. Furthermore, we prove that the classical Lie algebra $$sl_n(k)$$ , where char(k) does not divide n affords non-trivial faithful self-similarity.

We study the homological finiteness property FPm of permutational wreath products.

We study when a pro-p subdirect product S ⩽ G1 × . . . × Gn is of type FPm for m ⩾ 2 for some special pro-p groups Gi. In particular we treat the case when Gi is a finitely generated non-trivial free pro-p product different from C2 ∐ C2 if p = 2 or a non-abelian pro-p group from the class $\mathcal{L}$ defined in [12].

We study the growth of , where U is an open subgroup of and is a special class of pro-p groups defined in [7]. Furthermore for non-abelian we prove the core property: for pro-p subgroups such that H is finitely generated and N is non-trivial normal in G the index is always finite.

We classify the Bieri–Neumann–Strebel–Renz invariant Σ1(G) for a class of Artin groups based on the full graph with 4 vertices.

For an arbitrary sequence $(G_s)$ of subgroups of finite index in the generalised Thompson group $$F_{n, \infty} = \langle x_0, x_1, \ldots, x_m, \ldots \mid x_i^{x_j} = x_{i+ n-1} \hbox{ for } i > j \geq 0 \rangle$$ it is shown that $\sup_{s \geq 1} d(G_s) < \infty$ and that the deficiency gradient of $F_{n, \infty}$ with respect to $(G_s)$ is 0 provided $[G : G_s]$ tends to infinity. A higher dimensional analogue is considered for $n = 2$.

We consider some sufficient conditions for the pro-p completion of an orientable Poincare duality group of dimension n ≥ 3 to be a virtually pro-p Poincare duality group of dimension at most n − 2.

We develop new invariants similar to the Bieri-Strebel-Neumann-Renz invariants but in the category of Bredon modules (with respect to the class of the finite subgroups of G). We prove that for virtually soluble groups of type FP_{\infty} and finite extension of the Thompson group F the new invariants coincide with the classical ones.

We define an Artin group of circuit rank 1 as an Artin group based on a connected graph with fundamental group a free group of rank 1. For an Artin group G of circuit rank 1 the Bieri–Neumann–Renz–Strebel invariant Σ 1 ( G , ℤ ) ${\Sigma ^1(G, \mathbb {Z})}$ is calculated.

Let \({\mathcal {C}}\) be a class of finite groups. We study some sufficient conditions for the pro-\({\mathcal {C}}\) completion of an orientable \(\text{ PD }^3\)-pair over \(\mathbb {Z}\) to be an orientable profinite \(\text{ PD }^3\)-pair over \(\mathbb {F}_p\). More results are proven for the pro-\(p\) completion of \(\text{ PD }^3\)-pairs.

We classify the pro-p subdirect products of type FPm of Demushkin or free pro-p groups, in both cases of non-trivial Euler characteristic.

The Σ 2 -invariants of the generalised Thompson group F n , ∞ = 〈 x 0 , x 1 , … , x n , … | x i x j = x i + n − 1 for i > j ⩾ 0 〉 are calculated for n ⩾ 3 . The case n = 2 was solved in Bieri et al. (2010) [5] .

We show that for every finitely presented pro-$p$ nilpotent-by-abelian-by-finite group $G$ there is an upper bound on $\dim_{\mathbb{Q}_p} (H_1(M, \mathbb{Z}_p) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p )$, as $M$ runs through all pro-$p$ subgroups of finite index in $G$., Final accepted version. To appear in the Israel Journal of Mathematics

We show Lie algebra versions of some results on homological finiteness properties of subdirect products of groups. These results include a version of the 1-2-3 Theorem.

We study subdirect products of free and Demushkin pro-p groups of depth ∞ developing theory similar to the abstract case, see Bridson et al. (2002) [4] . Furthermore we classify when a subdirect product has homological type FP m for some m ⩾ 2 , a problem still open for abstract groups for m ⩾ 3 .

We prove that the pro-p completion of the orientable surface group of genus d is residually free pro-p if d is even or p = 2 . The case when p ≠ 2 , odd d > 1 is still open.

Let L be a finitely generated Lie algebra which is a split extension of a free nilpotent Lie algebra N by a finite dimensional abelian Lie algebra. Let V denote the quotient of N by its commutator subalgebra; we can regard V as a module for L/N. We discuss the relationship between the homological finiteness properties of V and those of L. In particular, we show that if L has type FPm and N has class c then V is 1 + c(m − 1)-tame (equivalently, the (1 + c(m − 1))th tensor power of V is finitely generated under the diagonal action of L/N).

Thompson's group F is the group of all increasing dyadic PL homeomorphisms of the closed unit interval. We compute † m .F / and † m .F I Z/, the homotopical and homological Bieri-Neumann-Strebel-Renz invariants of F , and show that † m .F / D † m .F I Z/ .A s an application, we show that, for every m, F has subgroups of type Fm� 1 which are not of type FPm (thus certainly not of type Fm).

We begin a study of a pro-p analogue of limit groups via extensions of centralizers and call \({\mathcal{L}}\) this new class of pro-p groups. We show that the pro-p groups of \({\mathcal{L}}\) have finite cohomological dimension, type FP∞ and non-positive Euler characteristic. Among the group theoretic properties it is proved that they are free-by-(torsion free nilpotent) and if non-abelian do not have a finitely generated non-trivial normal subgroup of infinite index. Furthermore it is shown that every 2 generated pro-p group in the class \({\mathcal{L}}\) is either free pro-p or abelian.

Let p be a prime number, T a class of finite groups closed under extensions, subgroups and quotients, and suppose that the cyclic group of order p is in T. We find some sufficient and necessary conditions for the pro-T completion of an abstract orientable Poincare duality group G of dimension 4 and Euler characteristic 0 to be a profinite orientable Poincare duality group of dimension 4 at the prime p with Euler p-characteristic 0. In particular we show that the pro-p completion y Gp of G is an orientable Poincare duality pro-p group of dimension 4 and Euler characteristic 0 if and only if G is p-good.

We prove that if G is a centre-by-metabelian pro-p group of type FPm over $\z_p$, so every homomorphic image of G is of type FPm over $\z_p$.

We establish some sufficient conditions for the profinite and pro- p p completions of an abstract group G G of type F P m FP_m (resp. of finite cohomological dimension, of finite Euler characteristic) to be of type F P m FP_m over the field F p \mathbb {F}_p for a fixed natural prime p p (resp. of finite cohomological p p -dimension, of finite Euler p p -characteristic). We apply our methods for orientable Poincaré duality groups G G of dimension 3 and show that the pro- p p completion G ^ p \widehat {G}_p of G G is a pro- p p Poincaré duality group of dimension 3 if and only if every subgroup of finite index in G ^ p \widehat {G}_p has deficiency 0 and G ^ p \widehat {G}_p is infinite. Furthermore if G ^ p \widehat {G}_p is infinite but not a Poincaré duality pro- p p group, then either there is a subgroup of finite index in G ^ p \widehat {G}_p of arbitrary large deficiency or G ^ p \widehat {G}_p is virtually Z p \mathbb {Z}_p . Finally we show that if every normal subgroup of finite index in G G has finite abelianization and the profinite completion G ^ \widehat {G} of G G has an infinite Sylow p p -subgroup, then G ^ \widehat {G} is a profinite Poincaré duality group of dimension 3 at the prime p p .

Let M be an irreducible, orientable, closed 3-manifold with fundamental group G. We show that if the pro-p completion \({\widehat{G}_{p}}\) of G is infinite then G is either soluble-by-finite or contains a free subgroup of rank 2.

Let G be a finitely generated group, and A a ℤ[G]-module of flat dimension n such that the homological invariant Σ n (G, A) is not empty. We show that A has projective dimension n as a ℤ[G]-module. In particular, if G is a group of homological dimension hd(G) = n such that the homological invariant Σ n (G, ℤ) is not empty, then G has cohomological dimension cd(G) = n. We show that if G is a finitely generated soluble group, the converse is true subject to taking a subgroup of finite index, i.e., the equality cd (G) = hd(G) implies that there is a subgroup H of finite index in G such that Σ∞(H, ℤ) ≠ ∅.

We show that an infinite cyclic covering space M' of a PD n -complex M is a PD n-1-complex if and only if ?(M) = 0, M' is homotopy equivalent to a complex with finite [(n-1)/2]-skeleton and p1(M') is finitely presentable. This is best possible in terms of minimal finiteness assumptions on the covering space. We give also a corresponding result for covering spaces M ? with covering group a PD r -group under a slightly stricter finiteness condition

Let G be a pro-p group of finite cohomological dimension and type FP ∞ and T be a finite p-group of automorphisms of G. We prove that the group of fixed points of T in G is again a pro-p group of type FP ∞ (in particular it is finitely presented). Moreover, we prove that a pro-p group G of type FP ∞ and finite virtual cohomological dimension has finitely many conjugacy classes of finite subgroups.

We study the injective endomorphisms φ of the Baumslag–Solitar group G(n,m)= 〈a,t | t-1 ant=am〉 for n, m ∈ ℤ∖{0} such that n, m ≠ ± 1, and show that φ (a) ∈ ∪ i ∈ ℤ∖{0} (ai)G. The proof is based on the Bass–Serre theory of graphs of groups.

We define and study the property finite presentability in the category  of Hopf algebras that are smash product of universal enveloping algebra of a Lie algebra by a group algebra. We show that for such Hopf algebras finite presentability is equivalent with finite presentability as an associative k-algebra.

We show that for a finitely generated group $G$ and for every discrete character $\chi\colon G \rightarrow \mathbb{Z}$ any matrix ring over the Novikov ring $\widehat{\mathbb{Z}G}_{\chi}$ is von Neumann finite. As a corollary we obtain that if $G$ is a non-trivial discrete group with a finite $K(G,1)$ CW-complex $Y$ of dimension $n$ and Euler characteristics zero and $N$ is a normal subgroup of $G$ of type $FP_{n-1}$ containing the commutator subgroup $G'$ and such that $G/N$ is cyclic-by-finite, then $N$ is of homological type $FP_n$ and $G/N$ has finite virtual cohomological dimension $$vcd(G/N) = cd(G) - cd(N).$$ This completes the proof of the Rapaport Strasser conjecture that for a knot-like group $G$ with a finitely generated commutator subgroup $G'$ the commutator subgroup $G'$ is always free and generalises an earlier work by the author where the case when $G'$ is residually finite was proved. Another corollary is that a finitely presentable group $G$ with $def(G) > 0$ and such that $G'$ is finitely generated and perfect can be only $\mathbb{Z}$ or $\mathbb{Z}^2$, a result conjectured by A. J. Berrick and J. Hillman in [1].

Let G be a profinite group, p a prime number and A a profinite Z ˆ 〚 G 〛 -module of finite projective p-dimension pd G , p ( A ) . Let N be a closed normal subgroup of G such that N is of finite cohomological p-dimension cd p ( N ) , H k ( N , F p ) is nonzero and finite for k = min { cd p ( N ) , pd G , p ( A ) } and N acts trivially on A. Then the virtual projective p-dimension vpd G / N , p ( A ) of A as a Z ˆ 〚 G / N 〛 -module is finite and vpd G / N , p ( A ) = pd G , p ( A ) − k . In the case when A = Z ˆ this generalizes the main result from [Th. Weigel, P.A. Zalesskii, Profinite groups of finite cohomological dimension, C. R. Acad. Sci. Paris Ser. I 338 (2004) 353–358].

The purpose of this paper is to try to unite some existing methods used in the classification results of metabelian Lie algebras and metabelian discrete groups of homological type FP2 via the language of Hopf algebras. This sheds more light on the similarities between the Lie and group cases and explains partially the differences. Still some of the results in the group case have homotopical flavour, using methods from covering spaces to establish that having homological type FP2 imposes strong condition on the first Σinvariant of the group ([4]). These methods do not have a purely algebraic counterpart. The Lie case was treated in [5, 6] with algebraic methods, and a Lie invariant (with a valuation flavour) for metabelian Lie algebras was proposed. This plays the same role in the Lie theory as the Bieri-Strebel Σ-invariant for metabelian groups. In this paper we do not suggest a new invariant but establish that the main result of [5] holds for some metabelian Hopf algebras. It is interesting to note that in both the Lie and group cases calculations with the second homology group of Abelian objects (Lie algebras or Abelian groups) viewed as modules over a commutative ring via the corresponding diagonal action was always quite helpful. The definition of the diagonal Lie and group actions can be united via the comultiplication map of Hopf algebras, and this was the starting point of our considerations. We study Hopf algebras H = U(L)#kG over a field k, that is, smash products of universal enveloping algebras U(L) of Lie algebras L over k by group rings kG, where G acts via conjugation on L and write X for the category of such Hopf algebras. This category is quite important. If char(k) = 0 it coincides with the category of cocommutative

Let G be a finitely presented pro- group with discrete relations. We prove that the kernel of an epimorphism of G to is topologically finitely generated if G does not contain a free pro- group of rank 2. In the case of pro-p groups the result is due to J. Wilson and E. Zelmanov and does not require that the relations are discrete ([15], [17]). For a pro-p group G of type FPm we define a homological invariant m(G) and prove that this invariant determines when a subgroup H of G that contains the commutator subgroup G is itself of type FPm. This generalises work of J. King for 1(G) in the case when G is metabelian [9]. Both parts of the paper are linked via two conjectures for finitely presented pro-p groups G without free non-cyclic pro-p subgroups. The conjectures suggest that the above conditions on G impose some restrictions on 1(G) and on the automorphism group of G

For G a locally compact group and i = 1 , 2 we define topological versions Σ top i ( G ) of the geometric homotopical invariants Σ 1 and Σ 2 of discrete groups. We calculate Σ top 1 ( G ) and Σ top 2 ( G ) for G = exp ( η ) ⋊ Q , η a nilpotent Lie algebra over a local p-adic field K and Q an abstract free abelian group of finite rank that acts on exp η via topological automorphisms. An important part of the structure of η is that it splits as a direct sum of one-dimensional (over K) K [ Q ] -modules. We conjecture the structure of the Bieri–Strebel–Renz invariant Σ 2 ( H ) for a discrete nilpotent-by-abelian S-arithmetic group H. The invariant Σ 2 ( H ) characterizes the finitely presented subgroups of H that contain the commutator.